Section 5.2 Volumes

DEFINITION OF VOLUME USING VERTICAL SLICES Let S be a solid that lies between x = a and x = b. If the cross-sectional area of S in the plane, through x and perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is

DEFINITION OF VOLUME USING HORIZONTAL SLICES Let S be a solid that lies between y = c and y = d. If the cross-sectional area of S in the plane, through y and perpendicular to the y-axis, is A(y), where A is a continuous function, then the volume of S is

SOME USEFUL AREAS 1.If the cross-section is a disk, we find the radius of the disk (in terms of x or y) and use A = π (radius) 2 2.If the cross-section is a washer, we find the inner radius and the outer radius of the washer (in terms of x or y) and use A = π (outer radius) 2 − π (inner radius) 2

VOLUME OF A DISK where R is the radius and w is the width

VOLUME OF A WASHER where R is the outside radius, r is the inside radius, and w is the width.

1.If the solid consists of adjacent vertical disks between x = a and x = b, we find the radius R(x) of the disk at x, and the volume is 2. If the solid consists of adjacent horizontal disks between y = c and x = d, we find the radius R(y) of the disk at y, and the volume is VOLUME BY DISKS

1.If the solid consists of adjacent vertical washers between x = a and x = b, we find the outside radius R(x) and inside radius r(x) of the washer at x, and the volume is 2.If the solid consists of adjacent horizontal washers between y = c and x = d, we find the outside radius R(y) and inside radius r(y) of the disk at y, and the volume is VOLUME BY WASHERS

PROCEDURE FOR FINDING VOLUMES BY THE METHOD OF DISKS 1.Sketch the region. Label intersection points. 2.Draw a slice and label the outside radius, inside radius (if necessary), and width. 3.Sum the volume of all disks (or washers); that is, set up and evaluate a definite integral.